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G = C62.31D4order 288 = 25·32

15th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.31D4, C23.8S32, (C2×C6).9D12, (C2×Dic3)⋊Dic3, (C6×Dic3)⋊1C4, C6.41(D6⋊C4), C625C41C2, C6.D41S3, C324(C23⋊C4), C62.34(C2×C4), (C22×C6).54D6, (C22×S3)⋊2Dic3, (C2×C62).7C22, C22.5(S3×Dic3), C2.11(D6⋊Dic3), C33(C23.6D6), C31(C23.7D6), C6.11(C6.D4), C22.3(D6⋊S3), C22.8(C3⋊D12), (S3×C2×C6)⋊1C4, (C2×C6).70(C4×S3), (C2×C3⋊D4).1S3, (C6×C3⋊D4).4C2, (C2×C6).6(C2×Dic3), (C3×C6.D4)⋊1C2, (C2×C6).12(C3⋊D4), (C3×C6).39(C22⋊C4), SmallGroup(288,228)

Series: Derived Chief Lower central Upper central

C1C62 — C62.31D4
C1C3C32C3×C6C62C2×C62C6×C3⋊D4 — C62.31D4
C32C3×C6C62 — C62.31D4
C1C2C23

Generators and relations for C62.31D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=a3c-1 >

Subgroups: 474 in 121 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×3], C22 [×3], C22 [×3], S3, C6 [×2], C6 [×12], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×6], C2×C6 [×9], C22⋊C4 [×2], C2×D4, C3×S3, C3×C6, C3×C6 [×3], C2×Dic3, C2×Dic3 [×5], C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C23⋊C4, C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×2], C62 [×3], C62, C6.D4, C6.D4 [×3], C3×C22⋊C4, C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4 [×2], C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.6D6, C23.7D6, C3×C6.D4, C625C4, C6×C3⋊D4, C62.31D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C23⋊C4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C23.6D6, C23.7D6, D6⋊Dic3, C62.31D4

Permutation representations of C62.31D4
On 24 points - transitive group 24T581
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 4 2 5 3 6)(7 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 22 5 19)(2 24 6 21)(3 20 4 23)(7 17)(8 13)(9 15)(10 14)(11 16)(12 18)
(1 17)(2 15)(3 13)(4 16)(5 14)(6 18)(7 19)(8 23)(9 21)(10 22)(11 20)(12 24)

G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,2,5,3,6)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,22,5,19)(2,24,6,21)(3,20,4,23)(7,17)(8,13)(9,15)(10,14)(11,16)(12,18), (1,17)(2,15)(3,13)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,2,5,3,6)(7,11,9,10,8,12)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,22,5,19)(2,24,6,21)(3,20,4,23)(7,17)(8,13)(9,15)(10,14)(11,16)(12,18), (1,17)(2,15)(3,13)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,4,2,5,3,6),(7,11,9,10,8,12),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,22,5,19),(2,24,6,21),(3,20,4,23),(7,17),(8,13),(9,15),(10,14),(11,16),(12,18)], [(1,17),(2,15),(3,13),(4,16),(5,14),(6,18),(7,19),(8,23),(9,21),(10,22),(11,20),(12,24)])

G:=TransitiveGroup(24,581);

On 24 points - transitive group 24T614
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 16 5 14 3 18)(2 17 6 15 4 13)(7 20 9 22 11 24)(8 21 10 23 12 19)
(2 13)(3 5)(4 17)(6 15)(7 10 22 19)(8 24 23 9)(11 12 20 21)(16 18)
(1 7)(2 23)(3 9)(4 19)(5 11)(6 21)(8 15)(10 17)(12 13)(14 22)(16 24)(18 20)

G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,5,14,3,18),(2,17,6,15,4,13),(7,20,9,22,11,24),(8,21,10,23,12,19)], [(2,13),(3,5),(4,17),(6,15),(7,10,22,19),(8,24,23,9),(11,12,20,21),(16,18)], [(1,7),(2,23),(3,9),(4,19),(5,11),(6,21),(8,15),(10,17),(12,13),(14,22),(16,24),(18,20)])

G:=TransitiveGroup(24,614);

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E6A···6F6G···6Q6R6S12A···12F
order122222333444446···66···66612···12
size112221222412121236362···24···4121212···12

39 irreducible representations

dim11111122222222244444444
type+++++++--++++--+
imageC1C2C2C2C4C4S3S3D4Dic3Dic3D6C4×S3D12C3⋊D4C23⋊C4S32S3×Dic3D6⋊S3C3⋊D12C23.6D6C23.7D6C62.31D4
kernelC62.31D4C3×C6.D4C625C4C6×C3⋊D4C6×Dic3S3×C2×C6C6.D4C2×C3⋊D4C62C2×Dic3C22×S3C22×C6C2×C6C2×C6C2×C6C32C23C22C22C22C3C3C1
# reps11112211211222611111224

Matrix representation of C62.31D4 in GL4(𝔽7) generated by

5232
1434
2233
0002
,
1526
0302
3301
0005
,
2052
4204
3331
1630
,
4145
2204
3424
4456
G:=sub<GL(4,GF(7))| [5,1,2,0,2,4,2,0,3,3,3,0,2,4,3,2],[1,0,3,0,5,3,3,0,2,0,0,0,6,2,1,5],[2,4,3,1,0,2,3,6,5,0,3,3,2,4,1,0],[4,2,3,4,1,2,4,4,4,0,2,5,5,4,4,6] >;

C62.31D4 in GAP, Magma, Sage, TeX

C_6^2._{31}D_4
% in TeX

G:=Group("C6^2.31D4");
// GroupNames label

G:=SmallGroup(288,228);
// by ID

G=gap.SmallGroup(288,228);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*c^-1>;
// generators/relations

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